An inverse problem in time-fractional diffusion equations with nonlinear boundary condition

2019 ◽  
Vol 60 (9) ◽  
pp. 091502
Author(s):  
Zhiyuan Li ◽  
Xing Cheng ◽  
Gongsheng Li
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations

scholarly journals Identification of points sources via time fractional diffusion equation

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6189-6201 ◽  
Author(s):  
A. Ghanmi ◽  
R. Mdimagh ◽  
I.B. Saad
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Stability Result ◽  
Fractional Diffusion ◽  
Cauchy Data ◽  
Diffusion Equations ◽  
Lipschitz Stability ◽  
Single Measurement ◽  
Fractional Diffusion Equations ◽  
Time Fractional Diffusion Equation

This article investigates the source identification in the fractional diffusion equations, by performing a single measurement of the Cauchy data on the accessible boundary. The main results of this work consist in giving an identifiability result and establishing a local Lipschitz stability result. To solve the inverse problem of identifying fractional sources from such observations, a non-iterative algebraical method based on the Reciprocity Gap functional is proposed.

Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

2012 ◽  
Vol 67 (8-9) ◽  
pp. 479-482 ◽  
Author(s):  
Junping Zhao
Keyword(s):  
Boundary Condition ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Blow Up ◽  
Existence Theorems ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Gradient Term

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut = div(a(u)b(x)▽u)+ f (x;u; |▽u|2; t) under nonlinear boundary condition ¶u=¶n+g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf’s maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-up rate. Our result indicates that the blow-up time T* may depend on a(u), while being independent of g(u) and f .

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